A variety of techniques have been used in determining the presence and in estimating quantities of hydrocarbons (oil and gas) in earth formations. These methods are designed to determine parameters of interest, including among other things, porosity, fluid content, and permeability of the earth formation surrounding a borehole drilled for recovering hydrocarbons. Typically, the tools designed to provide the desired information are used to log the borehole. Much of the logging is done after the borehole has been drilled. More recently, boreholes have been logged while drilling of the boreholes. This is referred to as measurement-while-drilling (“MWD”) or logging-while-drilling (“LWD”). Measurements have also been made when tripping a drillstring out of a borehole: this is called measurement-while-tripping (“MWT”).
The problem of diffusive motion under constraints of pore boundaries has many implications in a wide range of fields including biology, medicine, environmental science and the energy industry. NMR is an ideal candidate to study such a problem, but at high fields, the internal gradient due to magnetic susceptibility contrast becomes an important issue. In the case of sedimentary rocks, even low magnetic fields can generate a strong internal gradient that obscures the gradient of the external field. Numerous experimental studies as well as numerical calculations have been made to address this issue. For example, since the internal gradient is mainly produced by paramagnetic ions, paramagnetic doping and the effects of the density of paramagnetic ions have been studied. In addition, the distribution of internal gradients has been measured in sedimentary rocks. On the numerical side, the relaxation behavior has been calculated under various internal gradient and spatial restriction profiles. These studies on the one hand provide insight into the internal gradients present in porous materials, and on the other hand give an exact description of the relaxation behavior with ideal models. Practical issues, however, are often left unsolved due to the complexity of the natural pore space and the variation of the distribution of paramagnetic materials.
In this disclosure, it is shown that the complexity of the natural pore space and the variation in the distribution of paramagnetic materials can be simplified depending on the specific situation. One of the issues of particular interest is the Diffusion-T2 (D-T2) cross plot for fluid typing, which is used widely in well logging applications in the oil industry. It is easily postulated that the internal gradient effect causes an overestimate of diffusivity, but since modern logging tools employ multiple frequencies and associated gradients, and numerous sets of acquisition parameters, the exact influence of the internal gradient with restricted diffusion effect is unknown. A free diffusion equation with no regard to internal gradient is still generally used for data interpretation. The present disclosure uses a method that can incorporate internal gradient and restricted diffusion effects into the data inversion process. The simple model and associated calculation method discussed herein in detail constitute one embodiment of such a method.